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3. Can 200 be taken 4 times as well as 2 ? Can 2000, or any number, be taken 4 times ? Can 4, 40, or 400, be taken 2000 times ? Is it necessary, then, in multiplication, (as in addition and subtraction) that the factors should be of the same denomination or order ?
[In multiplication, it is usual to place the smaller factor at the right hand under the larger factor; but, as this is frequently impracticable in long calculations, and as the process is precisely the same, however placed, the student should never be hampered by any peculiar form, but, on the contrary, accustom himself to place the smaller factor in every possible situation,-above, below, at the right, left, or middle; and also in a horizontal position both before and after the larger, from which, in this case, it should be separated by the sign X or •. The latter will be necessary in the bills of parcels below. The student should also be cautioned against the awkward and unnecessary practice of ascertaining the various products on a separate part of his slate.]
4. Multiply 58369247 by each of the significant figures except 1, as a separate exercise, and prove by addition.
5. Multiply 30517284 by each significant figure, and prove. 6. Multiply 73261543 by each figure, and prove. 7. Multiply 85063472 by each figure, and prove. 8. Multiply 14759062 by each figure, and prove.
Exemplifications for the Black-board. When ciphers occur on the right hand of either factor, or of
both factors, or when decimal fractions occur in either or both factors. 9. Multiply 24500 by 50. 10. Multiply 3.76 by 4. 11. Multiply "24 by 4.
Solution by Suggestive Questions.-9th. The ciphers being neglected in both factors, what is 5 times 245 ? But, as wo
have multiplied by 5 in place of 50, how many times is the product too small? How can this be rectified ? See p. 115, 1. 8. But as 245 has been multiplied, instead of 245 hundred, how many times is the product still too small ? How can this be rectified? When ciphers at the right hand of factors are neglected, then, how many ciphers are requisite at the right of the product to rectify the process ? Ans. As many as there are in
10th. As the separatrix has been neglected in the smaller factor, thus making it 4 instead of 4 tenths, is the product too large or too small ? How many times ? How can this be rectified ? See p. 115, 1. 8. Rectify it, then. As the separatrix in the larger factor has also been neglected, thus making it 376 in place of 376, is the product too large or too small on this account? How many times? How can this be rectified ? Rectify it, then. Now, how many decimal places in the larger factor? In the smaller ? In both? In the corrected product? Is the number in the product the same as in both factors ? Had the number of fractional places in either or in both factors been less, what would have been the effect on the product ? What would have been the effect had there been more? Must the number of fractional places in the product, then, be always equal to the number in both factors ?
11th. But in this case only two figures occur, and yet three decimal places are wanting. What character is used when necessary to show the true place of figures? See
p. 114, 1. 24. Should it be placed to the right or left of the significant figures? See Oral Arithmetic, Chap. I., Sect. XV., p. 52, 1. 11.
From these three elucidations, then, may not the following be considered the ninth fundamental principle of arithmetic ?
IX. In multiplication, the number of ciphers at the right of
a product, or the number of decimal places which it contains, must always be made equal to the number of either in both factors.
12. Multiply 28700 by 40; by 600; by 90; by 300; as separate examples.
13. Multiply 25.7 by 6. How many decimal places should be in the product ?
14. Multiply 3658 by '8, previously determining the number of decimal places in this and in the examples below.
15. Multiply 46-42 by .005.
Escemplification for the Black-board.
20. Multiply 35264 by 14.
The long method. 35264
141056 Product of multiplicand by 4. 35264 Product of multiplicand by 1 and by 10 by position.
493696 Product of multiplicand by 14.
The short method.
Suggestive Questions. The Long Method.-How is the multiplicand multiplied by 10 by position ? See second Principle, p. 115. To which rank of the product by 4 is the first rank of the multiplicand added ? To which rank of the product by 4 is the second rank of the multiplicand added ? To which is the third ? &c. Could not these figures be added in without writing them over, and thus save two lines, or two thirds of the work? Let us try.
The Short Method. 4x4=16; 4X6=25 (24 and 1 carried)+4 (right hand figure of multiplicand)=29; 4x2=10 (8+2 carried)+6=16; 4X5=21 (20+1)+2=23; 4X3 =14 (12+2)+5=19; 1+3=4. [This short method only differs from the long method by adding in the right hand figure of the multiplicand as the work advances, instead of adding it in after the completion of the product of the multiplicand by the units' figure of the multiplier, as will readily be seen by a
comparison of the two methods above. Where the class is young, or dull, perhaps it might be proper to postpone the short process till the review. Great care should be taken, as usual, that the pupil does not use too many words; all that are necessary in the above exemplification of the short process are, sixteen; twenty-five twenty-nine; ten, sixteen; twenty-one, twenty-three; fourteen, nineteen; four. After a little practice, four out of these ten words might be dispensed with. [Which four ?]
21. Multiply 23542 by 11 to 19 severally, and prove by the long method ; or multiply by the long method, and prove by addition,
22. Multiply as above 4536249 by the numbers from 11 to 19 severally, and prove.
23. Multiply as above 49560-34 by 11 to 19 severally, and prove.
24. Multiply as above 7638-05 by 11 to 19 severally, and prove.
25. Multiply as above 697842:09 by 11 to 1.9 severally,
26. Multiply 32686534 as above by '11 to 19 severally, and prove.
27. Multiply 2343241, by short method, by 21 to 29 severally, and prove.
[This and the following exercises by the short method differ in nothing from the preceding, save that each figure of the multiplicand must be doubled, or trebled, or quadrupled, &c., according to the number of tens in the multiplier, before it is added to the product of the figure to its left. Every example should be proved by addition or by the long method.]
28. Multiply 7286159 by 21 to 29 severally, and prove. 29. Multiply 124932 by 31 to 39 severally, and prove. 30. Multiply 3946072 by 41 to 49 severally, and prove. 31. Multiply 2312412 by 51 to 59 severally, and prove. 32. Multiply 65749 by 61 to 69 severally, and prove. 33. Multiply 98357 by 71 to 79 severally, and prove. 34. Multiply 85679 by 81 to 89 severally, and prove. 35. Multiply 142312 by 91 to 99 severally, and prove. 36. Multiply 35241 by 324.
[This and the following exercises differ in nothing from the preceding, save that three or more products, in place of two, are added in mentally; that is, without being written down.]
Exemplifications for the Black-board.
324 Second factor.
140964 by 4
4 70482 by 2 and by 10 by position 105723 by 3 and by 100 by position =300
11418084 by 324.
The short method.
324 Second factor.
Suggestive Questions. The Long Method.-How is the second partial product multiplied by 10 by position ? See Second Principle, p. 115. How is the third partial product multiplied by 100 by position? Of what does the units' figure of the total product consist ? Ans. Of the product of the units of both factors. Of what two figures does the second rank of the total product consist; that is, of the product of 4 by what, and of 1 by what? [Point to the 4 and 1 as they are mentioned.] Of what three figures does the third rank of the total product consist; that is, of the product of 2 by what, of 4 by what, and of 1 by what? [Point to these figures.] Of what three figures does the fourth rank consist ? Of what three the fifth ? Of what three the sixth ? Of what does the seventh and eighth consist? Why could not these products be added in mentally, that is, without writing them during the progress of the work, and thus save nearly three fourths of the figures, as below?
The Short Method.-(4x1) 4; (4X4) 16+(2X1) 2=18; (4x2) 9 (1 carried)+(2X 4) 8+(3X1) 3=20; (4X5) 22 (2 carried) + (2x2) 4+ (3x4) 12=38; (4X3)=15+(2X5) 10+(3X2) 6=31; (2X3) 9+(3X5) 15=24; (3X3) 11. All the words necessary are the following: and very many even of these may be omitted after some practice. Four ; sixteen, eighteen ; nine, seventeen, twenty ; twenty-two, twentysix, thirty-eight; fifteen, twenty-five, thirty-one ; nine, twenty